A proximal alternating minimization algorithm for the largest C-eigenvalue of piezoelectric-type tensors

C-eigenvalues of piezoelectric-type tensors play an important role in piezoelectric effect and converse piezoelectric effect. While the largest C-eigenvalue of a given piezoelectric-type tensor has concrete physical meaning which determines the highest piezoelectric coupling constant. In this paper, we focus on computing the maximum C-eigenvalue of piezoelectric-type tensors which is a third degree polynomial problem. To do that, we first establish the equivalence between the proposed polynomial optimization problem (POP) and a multi-linear optimization problem (MOP) under conditions that the original objective function is concave. Then, an augmented POP (which can also be regarded as a regularized POP) is introduced for the purpose to guarantee the concavity of the underlying objective function. Theoretically, both the augmented POP and the original problem share the same optimal solutions when the compact sets are specified as unit spheres. By exploiting the multi-block structure of the resulting MOP, we accordingly propose a proximal alternating minimization algorithm to get an approximate optimal value of the maximum C-eigenvalue. Furthermore, convergence of the proposed algorithm is established under mild conditions. Finally, some preliminary computational results on synthetic data sets are reported to show the efficiency of the proposed algorithm.